Catenoid

A soap film spends as little area as its boundary allows. Between two equal rings, that economy takes the shape of a catenoid. Move the rings apart and the bridge answers by narrowing its waist.

It does not answer forever. This piece solves the stable branch of the catenoid boundary equation live. At a full ring separation of about 1.325 ring radii, the branch ends while the neck is still visibly there. Beyond it, there is no slightly thinner stable catenoid waiting next.

The equilibrium and its critical point are exact. Until Day 200 the collapse after that point was not: it was a clock, with the neck radius set to a chosen power of a number running from zero to one. It is now solved. The same curvature that defines the equilibrium keeps acting once no equilibrium exists, and that is the whole of the force. Nothing is seeded. Losing the solution is the driving.

Two things about the new collapse can be checked, and are. The surfaces the dynamics sit still at are exactly the catenoids this piece already draws, tested at four separations from a third of critical to ninety-seven per cent of it, with no measurable drift. And the pinch is a genuine finite-time singularity whose arrival time converges as the grid is refined, moving by less than three parts in ten thousand across a fourfold refinement.

One thing cannot be claimed. The rate at which the neck closes should approach a power law, and the exponent depends on what resists: two thirds if the inertia is a fluid, one if it is the film itself, and both follow from dimensions alone. The measured exponent climbs from 0.41 to 0.77 as the neck narrows and is still climbing at the finest grid this piece can afford. It is not converged, so no value is asserted. The first model tried here was a liquid column, which conserves volume and therefore rang about a fixed neck forever instead of collapsing. That failure is what identified the correct equilibrium. Film thickness, drainage, viscosity and the surrounding air remain outside the model, and the collapse is slowed by a constant factor, which changes the time axis and not the shape law.

Drag either ring to perform the experiment. After the snap, moving the rings back cannot restore the bridge. Re-dip them. Reversal repeats the distance; repair repeats the making.